At the heart of any education issue is the push from society at any given time in history. Mathematics has been no stranger to this societal pressure. A close look at the number of students taking algebra in high school for the years of 1909 to 1955 shows the number falling from 56.9% to 24.8% (Klein, 2003). A look at history offers some reasons for these falling numbers. The 1930’s saw a trend to the “Activity Movement” where teachers taught children and not subject matter. The 1940’s found the army having to teach math to its recruits so they could do bookkeeping or basic gunnery. The “Life Adjustment Movement” of the 1940s was taking place and students were to learn about consumer buying, insurance, and home budgeting, not algebra, geometry, and
trigonometry. Then in 1957, the launching of the Soviet satellite, Sputnik, embarrassed the United States and called attention to the lack of quality math and science education in the public schools. As a result, President Kennedy called on the nation to step up the rigor in math and science and put a man on the moon. Congress followed suit by enacting the 1958 National Defense Education Act to increase the number of math and science majors in colleges (Klein, 2003).
With the publishing of A Nation at Risk (1983) math shortcomings were called into public scrutiny. The report cited remedial math courses taught in public four year colleges had risen by 72% over the period of 1975 to 1980. During this same period, only 31% of high school graduates completed intermediate algebra. As the 1980’s came to a close the National Council of Teachers of Mathematics created the Curriculum and Evaluation Standards for School Mathematics, these standards were broken into grade
levels K-4, 5-8, and 9-12 but were met with some skepticism for not involving a connected progression of concepts for teachers to follow.
President Clinton had a part in math history by challenging public schools to do more with math. In his Call to Action for American Education in the 21st Century speech (1997), Clinton challenged math teachers saying, “…what 20% of our eighth-grade students learn in math is learned by most Japanese seventh-grade students” (Warren, 2008). In his speech then President Clinton claimed that as a nation we do not expect enough of our students and one of his charges was for every eighth-grade student to know algebra.
The 2000’s saw President Bush and his “leave no child behind” legislation with an emphasis on a new focus on math and science education by requiring a rigorous exam before graduation (Gill, 2004). President Obama followed suit during his September 2010 address stating that he had set a goal of moving America from the middle of the pack in science and math education to the top. To accomplish this President Obama, pledged to recruit 10,000 new math and science teachers over the next two years to support and strengthen our nation’s math and science education.
As a result of these political figures, we have seen a more focused effort on the part of national organizations such as, the National Council of Teachers of Math (NCTM), state education departments, and local school districts making efforts to strengthen math standards throughout K-12 education.
Contemporary Challenge Math Curriculum
In this study, two groups of sixth-grade students, those meeting the assessment requirements for placement and those not meeting the assessment requirement for
placement, but were placed due to parent or teacher recommendation, were placed in the same Challenge Math class--the grade level equivalent of seventh-grade math. With successful completion of Challenge Math, students would move to seventh-grade and take Pre-Algebra, and then Algebra in eighth-grade. Students taking these classes are taught concepts in four main standard areas; (a) number sense, (b)
geometric/measurement, (c) algebraic, and (d) data analysis/probability. The concern is whether sixth-grade students, particularly those placed based on teacher and/or parent pressure are ready to meet the conceptual and computational challenges required to learn and master this rigorous curriculum.
Number sense standard. In the number sense standard, students are to represent
and show relationships among rational numbers such as ordering rational numbers with fractions, decimals, and percents and demonstrate the meaning of arithmetic operations with positive fractions, decimals, and integers such as:
2/3 X 6 as two-thirds of six or 6 X 2/3 as six groups of two-thirds
Students are to compute fluently and accurately with appropriate strategies and tools and estimate and check reasonableness of answers using appropriate strategies and tools.
Student in the pre-algebra class studying the number sense standard will represent and show relationship among real numbers. These students will convert between
scientific notation and standard form including negative numbers. Proficiency with arithmetic operations with integers will be expected and the use of words and symbols will be used to explain properties such as the zero property of multiplication:
Students will investigate calculation of square integers, the square roots of perfect squares, and the square roots of whole numbers using technology. Problems involving ratios and percents will be solved such as:
x/5 = 10/17
As students study algebra this standard asks students to represent and show relationships among real numbers. Equivalent forms of irrational numbers are explored, such as:
√8 = 2√2
Students have to do more with arithmetic operations and numbers by demonstrating the meaning of these operations with real numbers. Students will investigate the effects of multiplication and division and computing positive powers and roots on the magnitude of quantities, for example:
If you take the square root of a number, will the result be smaller than the original? √1/4 = ½
Students will be able to multiply and divide numbers using scientific notation and simplify exponential expressions using powers. Students must be able to explain the method of computation when problem solving. Finally they need to be able to distinguish between relevant and irrelevant information in a given problem.
Geometric and Measurement. In the geometric/measurement standard students
are to compare and contrast properties and relationships of geometric shapes and objects. Students will learn how to use coordinate geometry to specify locations and describe relationships of those locations:
Students will use transformations and symmetry to analyze geometric shapes, use visualization to create geometric models in solving problems, and select and apply appropriate procedures, tools, and formulas to determine measurements.
In the pre-algebra geometric and measurement standard, students will be describing, comparing, and contrasting characteristics, properties, and relationships of geometric shapes and objects. Similar and congruent objects will be explored. The angles created by transversals dissecting parallel lines will be explored and understanding the relationship of the interior angles of a triangle will be examined. Students will use strategies to find the area and perimeter of complex shapes. And finally, the Pythagorean theorem will be used to find the missing lengths in right triangles and solve problems.
When investigating this standard, students in algebra will use coordinate
geometry to analyze and describe relationships in the coordinate plane. They will learn to apply slope to write and graph parallel and perpendicular lines. This standard also has students converting equivalent rates, such as:
Feet/second to miles/hour
Students will be able to apply units, systems, and formulas to solve problems.
Algebraic standard. The Challenge Math algebraic standard asks students to
represent and analyze relationships using algebraic symbols, such as: 2X = 6
Students will create, use, and interpret models of quantitative relationships:
Two times some number equals six or six less some number equals thirteen And, finally students can apply properties to solve equations and inequalities:
In the pre-algebra algebraic standard the focus is centered around beginning algebra concepts such as; describing relationships using algebraic expressions, equations and inequalities, identifying slope from tables and graphs, and determining rate of change from the slope of a line. An emphasis will be on graphing two variable equations using tables of ordered pairs and slope-intercept form. Students will graph linear inequalities and graphically solve linear systems of equations and inequalities. Evaluation of numerical expressions containing whole number exponents such as,
If x = 4, then (x + 3)2 + 5x = ?? will be central to this standard.
In the algebra course, the algebraic standard requires students to generalize, represent, and analyze linear, quadratic and exponential relationships using algebraic symbols. Students will use tables, graphs, and algebraic notation to convert among linear, quadratic, and exponential representations. Graphing and using ordered pairs to determine slope and intercepts of linear relationships from an equation or graph is a key objective to this standard. Students also need to model and analyze quantitative
relationships by using a variety of methods such as; graphs, tables, one variable
equalities, one variable inequalities, linear equations in slope intercept form, inequalities in slope intercept form, and system of linear equations with two variables. Another large concept in this standard is representing and solving equations and inequalities. Students should be able to simplify algebraic expressions involving exponents, such as:
(3x4)2= 3x4 X 3x4 or 9x8
Students should be able to multiply and divide a polynomial by a monomial: Divide, x4-5x3-2x by x2
This standard asks students to be able to solve quadratic equations by graphing, factoring, extracting the root, and quadratic formula. They should be able to multiply, divide, and simplify rational expressions, as well as, analyze and solve systems of two linear equations in two variables algebraically and graphically. Finally, students should simplify radical expressions and solve radical equations.
Data analysis and probability standard. Challenge math students in this
standard will find and interpret mean, median, mode, and range of data sets. They will tackle such concepts as explaining the difference between a population and a sample, selecting an appropriate measure of central tendency, evaluate predictions and inferences based on data, as well as, applying basic concepts of probability.
In this standard, pre-algebra students are asked to formulate questions that can be answered with data and then organize, display, and analyze the relevant data to answer their questions. Data will be presented in the form of circle graphs and box plots with and without technology. Central tendency along with quartiles for sets of data are explored. Along with the data analysis, students will explore the basic concepts of probability by computing the probabilities for independent compound events, dependent events and determining the odds of an event.
A key focus to this standard in algebra coursework is being able to formulate a question and design a survey, or an experiment, in which data is collected and displayed in a variety of formats, then select and use appropriate statistical methods to analyze the data. Students will interpret data represented by the normal distribution and formulate conclusions, as well as, explaining how sample size and transformations of data affect measure of central tendency. Students will develop and evaluate inferences to make
predictions and apply concepts of probability as they solve problems to answer their own questions.
National Council of Teachers of Math (NCTM) Standards
The United States does not have an official national math curriculum. In comparison with other industrialized and productive nations of the world, the United States relies on state and local control of the curriculum that is taught and assessed (Reys, Oscar, & Reys, 2003; Schmidt, Houang, & Cogan, 2002). This system can lead to an unfocused curriculum that fosters a culture of teaching what teachers feel like or what the textbook says to teach. Before 1985, no offering of a math national standard stating what should be taught and when it should be taught existed. The first year in which the NCTM standards arrived on the scene was 1989. These standards were updated in 2000, as a way to provide a focus for school leadership and teachers of math to key in on the necessary components of a sound mathematics curriculum (Reys, Chavez, & Reys, 2003).
The U.S. mathematics curriculum found in textbooks is characterized as a mile wide by an inch deep (Katz 2007; Schmidt & Cogan, 2009; Schmidt, McKnight, & Raizen, 1997). There is a growing consensus that while math textbooks typically cover lots of material few cover math concepts with substantial depth. In an effort to focus the math curriculum across the country the NCTM published updated standards in 2000. It is hoped that these standards will provide for learning goals for specific grade levels. These standards are organized in five content strands and five process strands (NCTM, 2002).
Content Strands
NCTM number and operations standard. According to the NCTM students
should be able to understand numbers, ways of representing numbers, relationships among numbers, and numbering systems. They should be able to understand meanings of operations and how they relate to one another. Finally students should compute fluently and make reasonable estimates. Within these three main objectives are a host of more specific learning objectives such as:
• Compare and order fractions, decimals, and percents to solve problems
(NCTM, 2002) • Develop an understanding of large numbers and recognize and
appropriately use exponential, scientific and calculator notation
• Use factors, multiples, prime factorization and relatively prime numbers to solve problems
• Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers
• Use the associative and commutative properties of addition and multiplication and distributive property over addition to simplify computations with integers, fractions and decimals
• Develop and analyze algorithms for computing with fractions, decimals and integers and develop fluency in their use
• Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent rates
NCTM algebra standard. In the algebra standard, students are to understand
patterns, relations, and functions. In this standard students are also expected to be able to represent and analyze mathematical situations and structures using algebraic symbols. Students should be able to use mathematical models to represent and understand
quantitative relationships and analyze change in various contexts. More specific learning goals will include items such as:
• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic rules
Super Chocolates are arranged in boxes so that a caramel is placed in the center of each array of four chocolates, as shown below. The dimensions of the box tell you how many columns and how many rows of chocolates come in the box. Develop a method to find the number of caramels in any box if you know its dimensions. Explain and justify your method using words, diagrams, or expressions.
(NCTM, 2002) • Identify functions as linear or nonlinear and contrast their properties from
tables, graphs, or equations
• Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope
• Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships
27 = 4x + 3 or y = 3x
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations
• Use graphs to analyze the nature of changes in quantities in linear relationships
NCTM geometry standard. Within the geometry standard, students will analyze
characteristics of properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships. Students will specify locations and describe spatial relationships using coordinate geometry and other representational systems. Another large concept area is applying transformations and using symmetry to analyze mathematical situations. Finally, students will use visualization, spatial,
reasoning, and geometric modeling to solve problems. More specific learning goals will include items such as:
• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
• Create and critique inductive and deductive arguments concerning
geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship
• Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides
Using slope from a coordinate plane to determine observations about a rhombus
and (NCTM, 2002)
• Examine the congruence, similarity, and line or rotational symmetry of objects using transformations
• Use two dimensional representations of three dimensional objects to visualize and solve problems such as those involving surface area and volume
• Use geometric models to represent, apply geometric relationships in areas outside mathematics to solve problems in everyday life
NCTM measurement standard. The measurement standard has students
understanding measurable attributes or objects and the units, systems, and processes of measurement. Applying appropriate techniques, tools, and formulas to determine
measurements is also a key component of this standard. More specific learning goals will include items such as:
• Understand relationships among units and convert from one unit to another within the same system and to other systems
• Understand, select, and use units of appropriate size and type to measure angles, perimeter, surface area, and volume
• Select and apply techniques and tools to accurately find length, area, volume, and angles to appropriate levels of precision
• Develop and use formulas to determine the circumference of circles, triangles, parallelograms, trapezoids, and circles and develop strategies to find area for more complex shapes
In (a) students could rearrange the trapezoid into a rectangle to learn that the formula to find area of a trapezoid is L x W or in (b) learn that a triangles area can be found by 1/2bh (NCTM, 2002). • Solve problems involving scale factors, using ratio and proportion • Solve problems involving rates and derived measurements for such
attributes as velocity and density
NCTM data analysis and probability standard. In the data analysis and
probability standard, students will formulate questions that can be addressed with data and collect, organize, and display relevant data to answer the questions. Students will be able to select and use appropriate statistical methods to analyze data. This standard asks students to develop and evaluate inferences and predictions that are based on data, as well as, understand and apply basic concepts of probability. Again underlying these broad concepts are more specific learning goals that include items such as:
• Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one
population
• Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem and leaf plots, box plots and scatterplots
• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit
• Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
Process Strands
NCTM problem solving standard. This is the first of the five process standards
within the NCTM’s curriculum. The NCTM feels that math instructional programs should incorporate problem solving in the curriculum from pre-kindergarten through 12th-grade. Students should be able to build new mathematical knowledge through problem solving, solve problems that arise in mathematics and in other contexts, apply and adapt a variety of appropriate strategies to solve problems, and monitor and reflect on